During the course of graduate school in theoretical physics, you tend to have a lot of conversations about things that get deep and philosophical, and at times your brain feels like it's going to explode, or you're reaching some kind of higher state of consciousness or having a psychedelic experience.

I can't count the number of times I felt this way in graduate school, but one of the most memorable times I can remember was when our theory group sat together one lunch and casually discussed a new paper which had just come out. I can't remember what the specific point was in the paper, but it hinged on what's known as the Boltzmann brain paradox. Some of us graduate students were relatively unfamiliar with the paradox, so the professors explained it to us.

Because entropy always increases as time goes forward, the early universe had very low entropy. Nobody knows what the first moment of the big bang looked like exactly. But presumably, it had zero entropy, or at least--lower entropy than any other moment in time after that first moment. This is a very special state, and it is usually accepted as the explanation for the "arrow of time", the fact that time looks different as it progresses forward from how it would look if you were to play everything backwards on rewind. If the universe had started out in a state of maximal entropy, where everything was homogenous and there were nothing but uniform randomness, going forwards or backwards would look the same... everything would just stay random-looking, no order would arise out of the random sea of chaos.

But instead, according to the standard cosmological picture of how the universe evolved, it started out in a very special, very unique, extremely low entropy state, sometimes called the initial singularity. That's all well and good, but there's one problem with it--out of all the possible states that the universe could go into, this one seems extremely special and extremely unlikely for it ever to get into in the first place. So our theories don't explain how this initial state got set up. What's worse, though, is that our theories do explain how random bubble universes could spontaneously fluxuate out of a background universe, appear for a moment, and then disappear back into nothingness, and you can calculate the probability for this to happen. The disturbing thing is, if you calculate the probability that one of these phantom bubble universes suddenly appears out of nowhere in the current state that we find ourselves in today, it's unlikely but nowhere near as unlikely as the scenario where the universe somehow managed to begin in the very special ultra low entropy state that it supposedly began in. The paradox is, that if you adhere to standard Bayesian probability theory and you ask yourself "how did I most likely get here, given my current perceptions and memories?" the answer--according to at least some relatively convincing recent scientific theories--is that probably, the universe didn't exist at all before a few moments ago, and all of your memories, and even all of the world outside a tiny little bubble around you doesn't exist. All of those memories seem to indicate that there was some past history before, but they could instead have just been conjured into existence out of nothingness only moments ago, and somehow it has convinced you to believe they actually happened. It could be that only our galaxy fluxuated into existence a moment ago, or only our solar system, only the earth--or as they explained it as we were sitting there, only the room we were sitting in! We sat there together, listening to the professors explain this, nodding our heads and asking questions. And I felt like we were in movie The Matrix or something, and I was being told that maybe--just maybe--the entire world I have known is all an illusion.

That's the Boltzmann brain paradox, I think in Boltzmann's time it had a slightly different form, but it has evolved over the decades as our understanding improves, and today it represents just as big a paradox as it once did, if anything moreso because the most persuasive theories of our time point more directly toward it being true. Nobody knows how to solve this paradox, so we just sort of waive away these undesirable solutions to the equations that involve these phantom bubble universes and assume that, despite any evidence to the contrary, our memories are reliable and the world we live in and the past we remember is real.

A key part of the mathematics behind the Boltzmann brain paradox that the professors explained to us that day is the Poincaré recurrence theorem. It says that if you start a system evolving according to any of the usual equations that define classical or quantum mechanics, and all of the particles start in a certain state, they will wander all around phase space for a long, long time, but eventually come back to that exact same state. You can even calculate the time it will take to return to the same state, and it's extremely long but it doesn't depend on what the initial state is. You could start the particles anywhere in phase space (I should explain what phase space is in the next part, I guess), and then they will eventually reach again the exact same state. But in between they will have wandered all around and had many adventures. This is called a Poincaré recurrence, and the time it takes for this to happen is called the Poincaré recurrence time. In order to calculate how unlikely it would be for the universe to get into the state that is usually thought of as the initial singularity, you just need to know the Poincaré recurrence time. It's basically how long it would take for all of the particles in the universe moving around randomly to just happen to wander into this unique state all at the same time.

This brings us to wandering sets. According to Wikipedia, if you have a set of points in a mathematical space that starts out somewhere, and if you have some rules that move those points all around the space, so that they follow some trajectories... and then if they eventually get back to where they started, they are called "non wandering". If, on the other hand, they never get back to where they started for all eternity, they are called "wandering". They just never make it home again. In physics, this is applied to a "phase space" which represents the possible states that the universe could be in at any given moment--I'll explain more about it in the next part. And the rules which move the points around are simply the laws of physics... they tell you how to take some initial conditions and time-evolve the system into a new state where all of the particles have moved somewhere else. In the new state they could have different positions or different velocities or both. But they will still be located somewhere in phase space, the space of all possible states they could be in. This distinction between spaces that contain wandering sets and those which don't turns out to be exactly the distinction needed to rigorously define what is meant by "dissipation". To be continued...

Comments

So is a wandering set is something like a irrational number, in that it never repeats (or in this case, loops)? Or is it just a matter of whether it is possible to reach the exact same positional arrangement again, regardless of direction, velocity, acceleration, etc.?

"So is a wandering set is something like a irrational number, in that it never repeats (or in this case, loops)?"

Yeah actually that's an excellent analogy I hadn't thought of! I think that's probably a really good way to think of them.

There may even be a more direct mathematical connection between irrational numbers and wandering sets, although I'd have to understand them a bit more to know what it is. I'll give that some thought and let you know if I come up with anything. But yes, the key is that it's a sequence that never repeats/loops.

There is one thing I glossed over though which may make this a little different... a point can be non-wandering if it just gets back to really really close to where it started, it doesn't necessarily have to come back to the exact same point, just the same "neighborhood", where neighborhoods have a precise mathematical definition.

Or is it just a matter of whether it is possible to reach the exact same positional arrangement again, regardless of direction, velocity, acceleration, etc.?

For wandering sets that are applicable to physics (not just pure mathematics), both the position and velocity would have to get back to what they started at (or very near to it), in order to count the point as non-wandering. I'm not sure about acceleration, I think in principle, the acceleration could be different the second time around but I'll have to think about that.

To follow up on the issue of acceleration... the known laws of physics In our universe happen to be second-order differential equations. For this case, knowing the position and velocity of all particles in the universe is enough to determine what their accelerations are. In other words, if you get back into a state where all of the positions and velocities are the same as some time before, then it's guaranteed that the accelerations will be the same too. (This is basically because acceleration is determined by all of the forces acting on a particle, which is in turn determined by the arrangement of other particles nearby)

Then the interesting question is--could you still define wandering sets in universes for which this wasn't the case. That I'm not sure of; mathematicians might be able to do it, but physicists wouldn't be very interested since it wouldn't apply to our world.